LINEAR AUTOMORPHISMS THAT ARE SYMPLECTOMORPHISMS

Abstract

Let K be the field of real or complex numbers. Let $(X \cong {\bf K}^{2n}, \omega)$ be a symplectic vector space and take $0 < k < n,N = ({2n \atop 2k})$. Let $L_1,\ldots,L_N \subset X$ be $2k$-dimensional linear subspaces which are in a sufficiently general position. It is shown that if $F : X \longrightarrow X$ is a linear automorphism which preserves the form $\omega^k$ on all subspaces $L_1,\ldots,L_N$, then $F$ is an $\epsilon_k$-symplectomorphism (that is, $F^*\omega = \epsilon_k\omega$, where $\epsilon_k^k = 1$). In particular, if ${\bf K} = \R$ and $k$ is odd then $F$ must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set ${\cal A}_{l,2r}$ of all $l$-dimensional linear subspaces on which the form $\omega$ has rank $\le 2r$ is linear in the Grassmannian $G(l,2n)$, that is, there is a linear subspace $L$ such that ${\cal A}_{l,2r} = L \cap G(l, 2n)$. In particular, the set ${\cal A}_{l,2r}$ can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.